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Detaylı Bibliyografya
Yazar:	Lee, Byoungwoo
Materyal Türü:	Recurso digital
Dil:
Baskı/Yayın Bilgisi:	Zenodo 2025
Konular:	Riemann zeta function large deviations log-correlated fields operator theory SAPZ framework mesoscopic statistics Gaussian free field Gaussian multiplicative chaos analytic number theory L-functions Hilbert–Schmidt operators trace-class operators
Online Erişim:	https://doi.org/10.5281/zenodo.17766422
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## Summary (v1.0) This record releases version v1.0 of a bridge paper between the SAPZ framework for the Riemann zeta function and the operator-generated log-correlated large deviation theory developed in a separate work ("Large deviation theory for operator-generated log-correlated fields"). The goal is deliberately intermediate: we work at the mesoscopic fluctuation level, identify the operator structure and the \(\dot H^{1/2}\) covariance, and formulate a conditional large deviation principle (LDP) for the SAPZ-regularized zeta log field, without entering the extreme-value or GMC regime. The paper is entirely deterministic and structural. No Fyodorov–Hiary–Keating (FHK) conjectures or extreme-value inputs are used. --- ## Overview On the number-theoretic side, the SAPZ program provides a robust regularization of \(\log|\zeta(1/2+it)|\) in large windows \([T,2T]\) via spectral smoothing, spacing–energy estimates, and zero-density inputs. On the probabilistic side, the operator-generated framework considers random trace-class operators \(A_T\) acting on a Hilbert space \(H\), and uses a Hilbert–Schmidt isometry \[ g \longmapsto G_g, \qquad g\in C_c^\infty(\mathbb{R}), \] to produce log-correlated fields \[ \langle X_T,g\rangle := \mathrm{Tr}(A_T G_g) \] with \(\dot H^{1/2}\) covariance structure. This paper constructs a mesoscopic SAPZ–zeta log field \[ L_T \in S_0'(\mathbb{R}), \] defined from a SAPZ-regularized version of \(\log|\zeta(1/2+it)|\) and a mesoscopic scaling \(w_T\), and shows that its covariance and finite-dimensional cumulants match those of an operator-generated log-correlated field at speed \(a_T\). Under additional SAPZ analytic assumptions, we obtain a conditional infinite-dimensional LDP on \(S_0'(\mathbb{R})\). --- ## Main results The main contributions can be summarized as follows. 1. **SAPZ–zeta log field construction.**      We define a SAPZ-regularized zeta log process    \[    \ell_T(t) = (K_T * \log|\zeta(1/2+i\cdot)|)(t)    \]    using SAPZ smoothing kernels \(K_T\) adapted to a mesoscopic scale \(w_T\), and from this we construct a random tempered distribution \(L_T\) on the rescaled coordinate \(u = (t-T)/w_T\):    \[    \langle L_T,g\rangle    = \int_{[T,2T]} \ell_T(t)\,g\!\left(\frac{t-T}{w_T}\right)\,\mathrm{d}t.    \] 2. **Operator representation and covariance bridge.**      Under SAPZ standing hypotheses (window separation, smoothing, and a spectral representation), we show that there exist random self-adjoint trace-class operators \(A_T\) on \(H=L^2(\mathbb{R})\) and a Hilbert–Schmidt isometry \(g\mapsto G_g\) such that    \[    \langle L_T,g\rangle = \mathrm{Tr}(A_T G_g) + R_T(g),    \]    with \(R_T(g)\) negligible at the fluctuation speed \(a_T \to \infty\). A covariance asymptotics hypothesis then yields    \[    \frac{1}{a_T} \,\mathrm{Cov}\bigl(\langle L_T,g\rangle,\langle L_T,h\rangle\bigr)    \longrightarrow    \int_{\mathbb{R}} |\xi|\,\widehat{g}(\xi)\,\overline{\widehat{h}(\xi)}\,\mathrm{d}\xi,    \]    i.e. the SAPZ–zeta field has leading-order covariance given by the \(\dot H^{1/2}\) inner product. 3. **Finite-dimensional cumulant control.**      We introduce SAPZ hypotheses on higher cumulants (for fixed test functions \(g_1,\dots,g_m\)) stating that all cumulants of order \(\ge 3\) are \(o(a_T)\), while the second-order cumulants scale like \(a_T\). Under these assumptions, the scaled log-moment generating function    \[    \Lambda_T^{(a)}(\theta)    := \frac{1}{a_T} \log \mathbb{E}\exp\Bigl(\sum_{j=1}^m \theta_j \langle L_T,g_j\rangle\Bigr)    \]    converges to the purely quadratic limit    \[    \Lambda(\theta)    = \frac12 \theta^\top Q \theta,    \qquad Q_{ij} = \langle g_i,g_j\rangle_{\dot H^{1/2}}.    \]    In particular, for each fixed finite family of test functions, the vector    \[    a_T^{-1/2} \bigl(\langle L_T,g_1\rangle,\dots,\langle L_T,g_m\rangle\bigr)    \]    is asymptotically Gaussian in the sense of cumulants. 4. **Conditional bridge to the operator LDP.**      We formulate strengthened SAPZ hypotheses (S1–S3), which lift the finite-dimensional cumulant control to a dense set of test functions, ensure Gärtner–Ellis type convergence of the scaled log-mgfs, and provide exponential moment bounds for Schwartz seminorms. Under these hypotheses, the auxiliary operator-generated field    \[    \langle X_T,g\rangle := \mathrm{Tr}(A_T G_g)    \]    satisfies the abstract operator LDP assumptions (cumulant scaling, log-mgf convergence, and exponential tightness), and hence its laws on \(S_0'(\mathbb{R})\) obey an LDP with speed \(a_T\) and good rate functional    \[    I(\varphi)    = \sup_{g\in C_c^\infty(\mathbb{R})}    \Bigl\{\langle \varphi,g\rangle - \tfrac12 \|g\|_{\dot H^{1/2}}^2\Bigr\}.    \]    Exponential equivalence between \(L_T\) and \(X_T\) then transfers the same LDP to the SAPZ–zeta field \((L_T)\). --- ## What is conditional in this paper The main theorem is explicitly conditional. The SAPZ standing hypotheses (H1–H5) and the bridge hypotheses (S1–S3) summarize the analytic tasks that must be verified in concrete SAPZ models: - multi-point correlation and spacing-energy estimates for zeta zeros sufficient to control higher cumulants; - uniform convergence of scaled log-mgfs for smeared statistics; - exponential moment bounds for a family of Schwartz seminorms controlling the topology of \(S_0'(\mathbb{R})\). Once these tasks are achieved, the full infinite-dimensional operator LDP for the SAPZ-regularized zeta log field follows from the abstract operator framework, without further zeta-specific fine-tuning. --- ## Relation to the SAPZ program and to previous work This paper sits between two larger pieces of work: - The SAPZ series on entropy–spectral regularization of \(L\)-functions, where smoothing kernels, zero-density inputs, and spacing-energy bounds are developed on the analytic number theory side. - The operator-generated log-correlated LDP paper, where a general LDP is proved for fields of the form \(\langle X_T,g\rangle = \mathrm{Tr}(A_T G_g)\) under structural cumulant and tightness assumptions. The present work shows how to: - rewrite the SAPZ-regularized zeta log field in trace form; - recover the canonical \(\dot H^{1/2}\) covariance from SAPZ-type hypotheses; - phrase the infinite-dimensional LDP for \(L_T\) as a purely operator-level problem, conditional on (S1–S3). We do **not** use extreme-value theory, Fyodorov–Hiary–Keating conjectures, or Gaussian multiplicative chaos. The focus is entirely on mesoscopic fluctuations and large deviations, not on maxima or critical chaos measures. --- ## Outlook The paper closes with several directions for future work: - constructing SAPZ-based multiplicative chaos measures for zeta and linking the LDP structure to extreme values; - extending the operator-generated framework to higher-dimensional log-correlated fields and to other families of \(L\)-functions; - formulating universality statements for SAPZ-regularized log fields across different arithmetic families, governed by the same operator LDP with \(\dot H^{1/2}\) rate functional. The SAPZ–Zeta Bridge is intended as a structural stepping stone: it separates the hard number-theoretic estimates from the abstract probabilistic machinery, and provides a clear list of analytic targets that, once resolved, yield a full large deviation principle for the mesoscopic zeta log field.

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